Bounds for Tree Automata with Polynomial Costs

We consider tree automata with costs over semirings in the sense of (Seidl, 1994). We de ne the concept of a nitely factorizing semiring and of a monotonic semiring, both as the generalization of well-known particular semirings, and show that the costniteness of tree automata with costs over nitely factorizing and monotonic semirings is decidable. We show that, for tree automata with costs over nitely factorizing and naturally ordered semirings, costniteness and boundedness are equivalent. Hence it is also decidable whether a tree automaton with costs over a nitely factorizing, monotonic, and naturally ordered semiring is bounded with respect to the natural order. With this we generalize the results of (Seidl, 1994) concerning the decidability of the boundedness of tree automata with costs over the classical semiring of natural numbers and the (max;+)-semiring of natural numbers.